**What is the Length of the Weighted Path of the Huffman Tree?**
**1. Tree Path Length**
The path length of a tree refers to the sum of the distances from the root node to each individual node in the tree. In a binary tree with the same number of nodes, a complete binary tree has the shortest path length.
**2. Weighted Path Length of the Tree (WPL)**
In certain applications, nodes in a tree may be assigned weights—real numbers that represent specific values or importance.
The weighted path length of a node is calculated by multiplying the distance from the node to the root (path length) by its weight. The overall **Weighted Path Length (WPL)** of the tree is the sum of these weighted path lengths for all **leaf nodes**. It is typically represented as:
$$
\text{WPL} = \sum_{i=1}^{n} w_i \times l_i
$$
Where:
- $ n $: the number of leaf nodes
- $ w_i $: the weight of the $ i^{th} $ leaf node
- $ l_i $: the path length from the root to the $ i^{th} $ leaf node
This value is also known as the **cost of the tree**.
**3. Optimal Binary Tree or Huffman Tree**
Among all possible binary trees that can be constructed using $ n $ leaf nodes with given weights $ w_1, w_2, ..., w_n $, the one with the **smallest weighted path length** is called the **optimal binary tree**, or **Huffman tree**.
For example, suppose we have four leaf nodes with weights 7, 5, 2, and 4. Different configurations yield different WPLs:
- (a) WPL = 7×2 + 5×2 + 2×2 + 4×2 = 36
- (b) WPL = 7×3 + 5×3 + 2×1 + 4×2 = 46
- (c) WPL = 7×1 + 5×2 + 2×3 + 4×3 = 35
Among these, configuration (c) gives the smallest WPL, making it the **Huffman tree**.
**Notes:**
1. When all leaf weights are equal, a complete binary tree is always the optimal one. Otherwise, it may not be.
2. In an optimal binary tree, nodes with larger weights tend to be closer to the root.
3. There may be multiple valid Huffman trees, but all will have the **same minimum WPL**.
**How to Calculate the Weighted Path Length of a Huffman Tree**
**Problem Description**
You are given two lines of input. The first line contains a positive integer indicating the number of leaf nodes. The second line contains a list of positive integers representing the weights of the leaf nodes. Your task is to construct a Huffman tree and calculate its weighted path length (WPL).
**Input Format**
- The first line contains an integer $ n $ (number of leaf nodes).
- The second line contains $ n $ space-separated integers representing the weights of the leaf nodes.
**Output Format**
- Output the calculated **weighted path length (WPL)** of the Huffman tree.
**Sample Input**
5
4 5 6 7 8
**Sample Output**
69
**About the Huffman Tree**
**1. Path Length**
A path in a tree is formed by traversing from one node to another. The number of edges along this path is called the **path length**.
**2. Tree Path Length**
The total path length of a tree is the sum of the path lengths from the root to every node in the tree.
**3. How to Build a Huffman Tree**
To build a Huffman tree:
- Create a priority queue (min-heap) containing all the leaf nodes.
- While there is more than one node in the queue:
- Remove the two nodes with the smallest weights.
- Create a new internal node with a weight equal to the sum of the two.
- Add this new node back into the queue.
- The final node in the queue is the root of the Huffman tree.
Once the tree is built, the WPL can be calculated by summing up $ w_i \times l_i $ for all leaf nodes. This ensures the most efficient encoding in terms of minimizing the average code length.
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